Total Variation (TV) is a widely-used measure for intensity continuity of images. It has been applied in many applications such as image restoration, deconvolution, decompression, inpainting, etc.
For instance T. Chan and C. Wong, “Total Variation Blind Deconvolution”, IEEE Transactions on Image Processing, 7(3), 370-375 (1998), describe use for blind deconvolution and F. Guichard and F. Malgouyres, “Total Variation based interpolation”, Proc. European Signal Processing Conf., 3, 1741-1744 (1998), use for resolution enhancement. Another use case is decompression described by F. Alter, S. Durand, J. Froment, in “Adapted Total Variation for Artifact Free Decompression of JPEG Images”, J. Math. Imaging and Vision, 23(2), 199-211 (2005).
In particular, TV denoising is remarkably effective at simultaneously preserving edges while removing noise in flat regions, which is a significant advantage over the intuitive techniques such as linear smoothing or median filtering. The idea is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the absolute gradient of the signal is high.
According to this principle, reducing the total variation of the signal subject to it being a close match to the original signal, removes unwanted detail whilst preserving important details such as edges.
Typically, TV is calculated with the horizontally and vertically gradient images. Denote an image by I, its horizontally and vertically gradient images, ∇xI and ∇yI are defined as∇xI=I(x+1,y)−I(x,y) and ∇yI=I(x,y+1)−I(x,y).
Then TV is calculated, wherein sqrt(•) calculates the square root of its argument, by:TV(I)=Σi,jsqrt(∇xI(i,j)2+∇xI(i,j)2) or   (1)TV(I)=Σi,j(|∇xI(i,j)|+|∇xI(i,j)|)  (2)
Classical TV denoising tries to minimizes the Rudin-Osher-Fatemi (ROF) denoising model:minfTV(f)+λ*(∥f−n∥2)2/2  (3)where n is the noisy image, TV(f) is the total variation of f, and λ is a parameter which controls the denoising intensity.
The idea of TV regularization is increasingly employed in compressive sensing recently. For instance, it is proposed to recover images by a few samples based on the following equation, where Φ is some certain sampling matrix and y is the obtained sample.minfTV(f)+λ*(∥y−Φf∥2)2/2  (4)
As can be seen in Eq. (3), traditional TV regularization does not consider the content of images, it simply smoothes the entire image with equivalent intensity from both horizontal and vertical direction. Therefore, the edges are smoothed more or less after TV denoising, especially the oblique edges. As a conclusion, the gradients along horizontal and vertical direction are not robust enough for various images. X. Shu and N. Ahuja, “Hybrid Compressive Sampling via a New Total Variation TVL1”, Proc. ECCV'10, 393-404 (2010), propose a so called TV11 for compressive sampling. TV11 calculation is based on the horizontal and vertical gradients, and in addition, two diagonal partial gradients, ∇x∇yI(i,j) and ∇y∇xI(i,j) to enforce the diagonal intensity continuity.